Lilavati (Bhaskaracharya)

The sum of these products is 240. When divided by the total sum of the gold weights, which is 20, the resulting average fineness original: "kanaka-kavali", the calculated purity of the combined mass is 12.

If these pieces of gold are refined until they weigh only 16 Māṣas: a unit of weight, then the fineness becomes 15. Conversely, if the gold is refined to reach a fineness of 16, then the weight becomes 15 māṣas.

The reasoning for this is: the product of the total gold weight and the resulting fineness [is the total "value" of the gold]. To find an unknown fineness, subtract the known products from this total and divide by the weight of the gold with the unknown fineness.

Verse 14:
Tell the measure of the unknown fineness.

Statement:
[With weights and finenesses of] 5, 20, 6... the result is gold of 12 fineness. The obtained value for the unknown fineness is 15.

Formula for Finding the Unknown Weight

Verse 15:
The total weight of the gold multiplied by the resulting fineness, minus the sum of the products of the known weights and their respective finenesses, and then divided by the fineness of the unknown gold piece—this, the experts say, will be the weight of that unknown piece.

Example:
There is a piece of gold with a fineness of 11 original: "daśendu", literally "ten-one," a poetic way to say 11 and a weight of 1 Māṣa original: "chandra", literally "moon," signifying the number 1. To this, some gold of 16 fineness is added. When they are melted together, the resulting mixture has a fineness of 12. Tell me, O seeker, how many māṣas of the 16-fineness gold were there?

Statement:
10, 15, 16 These figures represent the values to be manipulated in the calculation.
The resulting fineness after melting is 12.
The weight is thus obtained.